A variant \(I^{\prime }(G)\)  of the incidence matrix of \(G\)  is presented in
[1]}. Orient the edges of \(G\)  by assigning a direction to each
one. Then if \(\mathfrak {e}=\left[ a,b\right] \)  with the direction of \(\mathfrak {e}\)  being \(a\longrightarrow b\) , change the 1 in the \(\left( a,\mathfrak {e}\right) \)  position of \(I(G)\)  to \(-1\)  . Proposition 4.3 of [1]} now shows that \(\mathrm {rank}I^{\prime }(G)=p-k(G)\) , with \(k(G)\)  the
number of connected components of \(G\) .
